Multilevel polynomial regression analysis (MPRA; see Nestler et al. (2019)) is an adaptation of
polynomial regression analysis (PRA; see Edwards
& Parry (1993)) that can be used when data have a multilevel
structure (e.g., observations across multiple time points nested within
participants). In the current study, MPRA allows testing and
interpreting both the linear effects of vitality and learning
separately, as well as the effects of congruent vitality and learning
scores using data collected at four time points with a one-month time
lag between the assessments. In the following, we use the index \({m}\) to denote the Level 2 unit (\(m = 1…N\)). To assess the existence of
congruence effects, the following parameters are estimated at the
population level: the slope of the line of congruence (LOC) (i.e.,
vitality = learning) as related to the outcome variable is given by
\(\alpha_{1m} = b_{1} + b_{2}\). Here,
\(b_{1}\) is the unstandardized
regression coefficient for the (latent) vitality variable and \(b_2\) is the regression coefficient for the
(latent) learning variable. The curvature along the line of perfect
agreement is specified by \(\alpha_{1m} =
b_{3} + b_{4} + b_{5}\). Here, \(b_{3}\) is the regression coefficient for
(latent) vitality squared, b4 is the regression coefficient for the
cross-product of the (latent) vitality and the (latent) learning
variable, and b5 is the unstandardized regression coefficient for latent
learning squared. The slope of the line of incongruence (LOIC) is
defined as \(\alpha_{3m} = b_{1} -
b_{2}\). The curvature of the line of incongruent vitality and
learning levels as related to the outcome variable is specified as \(\alpha_{4m} = b_{3} - b_{4} + b_{5}\). Each
of the regression coefficients \(b_{1}\) to \(b_{5}\) includes a fixed effect and Level 2
residual terms. The fixed effects of the regression coefficients are
used to estimate the average response surface parameters \(\hat{\alpha}_{1}\) to \(\hat{\alpha}_{5}\) (Nestler et al., 2019).
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References
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The dots represent the raw data points. The x and y axes represent the person-mean centred vitality and learning scores, respectively.
The dots represent the raw data points. The x and y axes represent the person-mean centred vitality and learning scores, respectively.
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| Model test | \(\chi^2\) | \(\textit{p}\) | CFI | TLI | RMSEA | SRMR | \(\Delta\chi^2\) | \(\Delta\)CFI | \(\Delta\)TLI | \(\Delta\)RMSEA | \(\Delta\)SRMR |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Configural invariance | 404.625 (188) | <.001 | 0.990 | 0.985 | 0.033 | 0.032 | - | - | - | - | - |
| Weak invariance | 412.586 (200) | <.001 | 0.990 | 0.986 | 0.032 | 0.032 | 7.961 (12) | <0.001 | 0.001 | -0.001 | <0.001 |
| Strong invariance | 434.928 (212) | <.001 | 0.990 | 0.986 | 0.031 | 0.032 | 22.342 (12) | <0.001 | <0.001 | <0.001 | <0.001 |
| Strict invariance | 445.956 (218) | <.001 | 0.989 | 0.986 | 0.031 | 0.033 | 11.028 (6) | <0.001 | <0.001 | <0.001 | <0.001 |
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Lagged multilevel model including the effects of predictors measured at t-1 on physical health measured at t
| Model 1 | Model 2 | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Predictors | b | SE | beta | SE beta | p | b | SE | beta | SE beta | p | std. p |
| (Intercept) | 5.40 | 0.17 | -0.00 | 0.03 | <0.001 | 5.62 | 0.29 | 0.00 | 0.03 | <0.001 | 0.967 |
| T-1.Physical.health | -0.25 | 0.03 | -0.12 | 0.01 | <0.001 | -0.25 | 0.03 | -0.12 | 0.01 | <0.001 | <0.001 |
| Thriving | 0.01 | 0.04 | 0.00 | 0.02 | 0.766 | ||||||
| Vitality | -0.01 | 0.03 | -0.01 | 0.02 | 0.734 | 0.734 | |||||
| Learning | 0.02 | 0.03 | 0.01 | 0.01 | 0.350 | 0.350 | |||||
| Vitality squared | -0.02 | 0.02 | -0.01 | 0.02 | 0.459 | 0.459 | |||||
| Vitality x learning | 0.00 | 0.02 | 0.00 | 0.02 | 0.855 | 0.855 | |||||
| Learning squared | 0.02 | 0.02 | 0.02 | 0.02 | 0.319 | 0.319 | |||||
| Positive affect | -0.01 | 0.04 | -0.01 | 0.02 | 0.704 | -0.01 | 0.04 | -0.00 | 0.02 | 0.780 | 0.780 |
| Negative affect | -0.02 | 0.03 | -0.01 | 0.01 | 0.582 | -0.02 | 0.03 | -0.01 | 0.01 | 0.596 | 0.596 |
| Demands | -0.00 | 0.03 | -0.00 | 0.01 | 0.971 | -0.00 | 0.03 | -0.00 | 0.01 | 0.979 | 0.979 |
| Autonomy | 0.02 | 0.03 | 0.01 | 0.01 | 0.425 | 0.02 | 0.03 | 0.01 | 0.01 | 0.428 | 0.428 |
| Coworker support | -0.01 | 0.03 | -0.01 | 0.02 | 0.638 | -0.01 | 0.03 | -0.01 | 0.02 | 0.702 | 0.702 |
| Supervisor support | 0.00 | 0.03 | 0.00 | 0.02 | 0.914 | -0.00 | 0.03 | -0.00 | 0.02 | 0.997 | 0.997 |
| Time | 0.01 | 0.01 | 0.01 | 0.01 | 0.470 | 0.01 | 0.01 | 0.01 | 0.01 | 0.503 | 0.503 |
| Mean vitality | -0.30 | 0.16 | -0.32 | 0.17 | 0.063 | 0.063 | |||||
| Mean learning | 0.14 | 0.17 | 0.14 | 0.17 | 0.409 | 0.409 | |||||
| Mean vitality squared | 0.06 | 0.03 | 0.41 | 0.20 | 0.038 | 0.038 | |||||
| Mean vitality x learning | -0.02 | 0.05 | -0.09 | 0.25 | 0.713 | 0.713 | |||||
| Mean learning squared | -0.02 | 0.03 | -0.12 | 0.20 | 0.566 | 0.566 | |||||
| Mean thriving | -0.01 | 0.06 | -0.01 | 0.05 | 0.853 | ||||||
| Mean positive affect | 0.01 | 0.06 | 0.01 | 0.06 | 0.883 | -0.01 | 0.07 | -0.01 | 0.06 | 0.920 | 0.920 |
| Mean negative affect | -0.31 | 0.04 | -0.26 | 0.03 | <0.001 | -0.30 | 0.04 | -0.26 | 0.03 | <0.001 | <0.001 |
| Mean demands | -0.06 | 0.03 | -0.06 | 0.03 | 0.046 | -0.06 | 0.03 | -0.05 | 0.03 | 0.075 | 0.075 |
| Mean autonomy | 0.07 | 0.03 | 0.07 | 0.03 | 0.047 | 0.06 | 0.03 | 0.06 | 0.03 | 0.088 | 0.088 |
| Mean coworker support | 0.01 | 0.04 | 0.01 | 0.04 | 0.723 | 0.01 | 0.04 | 0.01 | 0.04 | 0.777 | 0.777 |
| Mean supervisor support | 0.01 | 0.04 | 0.01 | 0.04 | 0.823 | 0.01 | 0.04 | 0.01 | 0.04 | 0.859 | 0.859 |
| Age | -0.01 | 0.00 | -0.16 | 0.03 | <0.001 | -0.01 | 0.00 | -0.17 | 0.03 | <0.001 | <0.001 |
| Gender | -0.03 | 0.05 | -0.02 | 0.03 | 0.492 | -0.04 | 0.05 | -0.02 | 0.03 | 0.435 | 0.435 |
| Education | 0.08 | 0.03 | 0.08 | 0.03 | 0.006 | 0.08 | 0.03 | 0.08 | 0.03 | 0.007 | 0.007 |
| Organizational tenure | 0.01 | 0.00 | 0.06 | 0.03 | 0.073 | 0.01 | 0.00 | 0.07 | 0.03 | 0.048 | 0.048 |
| Random Effects | |||||||||||
| σ2 | 0.256 | 0.250 | |||||||||
| τ00 | 0.398 Person.ID | 0.397 Person.ID | |||||||||
| τ11 | 0.025 Person.ID.Thriving | 0.039 Person.ID.Vitality | |||||||||
| ρ01 | -0.118 Person.ID | -0.033 Person.ID | |||||||||
| ICC | 0.611 | 0.618 | |||||||||
| N | 888 Person.ID | 888 Person.ID | |||||||||
| Observations | 2136 | 2136 | |||||||||
| Marginal R2 / Conditional R2 | 0.123 / 0.658 | 0.128 / 0.667 | |||||||||
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Lagged multilevel model including the effects of predictors measured at t-1 on mental health measured at t
| Model 1 | Model 2 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Predictors | b | SE | beta | SE beta | p | b | SE | beta | SE beta | p |
| (Intercept) | 4.39 | 0.12 | 0.00 | 0.02 | <0.001 | 4.14 | 0.20 | 0.00 | 0.02 | <0.001 |
| T-1.Mental.health | -0.37 | 0.03 | -0.16 | 0.01 | <0.001 | -0.37 | 0.03 | -0.17 | 0.01 | <0.001 |
| Thriving | 0.05 | 0.04 | 0.02 | 0.02 | 0.154 | |||||
| Vitality | 0.00 | 0.03 | 0.00 | 0.02 | 0.941 | |||||
| Learning | 0.06 | 0.03 | 0.03 | 0.01 | 0.020 | |||||
| Vitality squared | -0.03 | 0.02 | -0.02 | 0.01 | 0.191 | |||||
| Vitality x learning | 0.02 | 0.02 | 0.01 | 0.02 | 0.361 | |||||
| Learning squared | 0.01 | 0.02 | 0.01 | 0.01 | 0.577 | |||||
| Positive affect | -0.00 | 0.04 | -0.00 | 0.01 | 0.947 | 0.00 | 0.04 | 0.00 | 0.01 | 0.985 |
| Negative affect | -0.05 | 0.03 | -0.02 | 0.01 | 0.188 | -0.05 | 0.03 | -0.02 | 0.01 | 0.166 |
| Demands | -0.03 | 0.03 | -0.01 | 0.01 | 0.270 | -0.03 | 0.03 | -0.01 | 0.01 | 0.271 |
| Autonomy | 0.01 | 0.03 | 0.00 | 0.01 | 0.749 | 0.01 | 0.03 | 0.01 | 0.01 | 0.619 |
| Coworker support | 0.02 | 0.03 | 0.01 | 0.01 | 0.586 | 0.01 | 0.03 | 0.01 | 0.01 | 0.664 |
| Supervisor support | -0.02 | 0.03 | -0.01 | 0.01 | 0.500 | -0.02 | 0.03 | -0.01 | 0.01 | 0.519 |
| Time | 0.00 | 0.01 | 0.00 | 0.01 | 0.752 | 0.01 | 0.01 | 0.00 | 0.01 | 0.695 |
| Mean vitality | 0.61 | 0.11 | 0.56 | 0.10 | <0.001 | |||||
| Mean learning | -0.30 | 0.12 | -0.26 | 0.10 | 0.012 | |||||
| Mean vitality squared | -0.06 | 0.02 | -0.34 | 0.12 | 0.005 | |||||
| Mean vitality x learning | -0.02 | 0.03 | -0.11 | 0.15 | 0.463 | |||||
| Mean learning squared | 0.05 | 0.02 | 0.27 | 0.13 | 0.032 | |||||
| Mean thriving | 0.03 | 0.04 | 0.03 | 0.03 | 0.440 | |||||
| Mean positive affect | 0.42 | 0.04 | 0.35 | 0.03 | <0.001 | 0.30 | 0.05 | 0.25 | 0.04 | <0.001 |
| Mean negative affect | -0.67 | 0.03 | -0.50 | 0.02 | <0.001 | -0.65 | 0.03 | -0.48 | 0.02 | <0.001 |
| Mean demands | -0.02 | 0.02 | -0.01 | 0.02 | 0.452 | -0.00 | 0.02 | -0.00 | 0.02 | 0.983 |
| Mean autonomy | 0.01 | 0.02 | 0.01 | 0.02 | 0.576 | 0.03 | 0.02 | 0.02 | 0.02 | 0.268 |
| Mean coworker support | 0.07 | 0.03 | 0.06 | 0.02 | 0.021 | 0.06 | 0.03 | 0.05 | 0.02 | 0.025 |
| Mean supervisor support | -0.06 | 0.03 | -0.05 | 0.03 | 0.031 | -0.07 | 0.03 | -0.07 | 0.02 | 0.006 |
| Age | 0.01 | 0.00 | 0.06 | 0.02 | 0.003 | 0.01 | 0.00 | 0.06 | 0.02 | 0.003 |
| Gender | -0.01 | 0.04 | -0.01 | 0.02 | 0.732 | -0.01 | 0.03 | -0.00 | 0.02 | 0.838 |
| Education | -0.04 | 0.02 | -0.04 | 0.02 | 0.037 | -0.04 | 0.02 | -0.03 | 0.02 | 0.042 |
| Organizational tenure | -0.00 | 0.00 | -0.01 | 0.02 | 0.744 | -0.00 | 0.00 | -0.01 | 0.02 | 0.525 |
| Random Effects | ||||||||||
| σ2 | 0.266 | 0.273 | ||||||||
| τ00 | 0.138 Person.ID | 0.123 Person.ID | ||||||||
| τ11 | 0.079 Person.ID.Thriving | 0.036 Person.ID.Vitality | ||||||||
| ρ01 | -0.053 Person.ID | 0.017 Person.ID | ||||||||
| ICC | 0.360 | 0.323 | ||||||||
| N | 888 Person.ID | 888 Person.ID | ||||||||
| Observations | 2136 | 2136 | ||||||||
| Marginal R2 / Conditional R2 | 0.580 / 0.731 | 0.591 / 0.723 | ||||||||